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Understanding the Gambler’s Fallacy: The Fallacy of the Maturity of Chances

Past results of random independent events, like a coin flip, don’t affect future results. The mistaken belief that past results affect future results is known as “the Gambler’s Fallacy” (AKA the Fallacy of the Maturity of Chances, or the Monte Carlo Fallacy).

To phrase this another way, past events (that are random and independent) don’t make a future result of a subsequent event more or less likely.

For example, a “fair” coin always has a 50/50 chance of heads or tails, regardless of past results.[1][2]

No matter how many times a “fair coin” lands on heads or tails in a row, the chances it will land on heads or tails next is always exactly 1 in 2, 50%, 1:1, etc. Nothing can change this, each independent and random result is… random and independent.

The same is true for any game of chance, nothing that happened in the past affects the current odds, nothing is ever due, chances do not “mature”, every unique random event retains its probability, the odds of a fair coin landing on heads is always 1:1… even if it just landed on heads or tails ten times in a row.

Below we use different examples of chance-based games (as opposed to skill-based) to explore the ins-and-outs of the persistent gambler’s fallacy myth.

TIP: Before any flips are made, the odds of a “fair coin” landing on heads every time gets less-and-less likely the more flips we consider. Despite this, each flip itself has a 50% chance of landing on heads or tails. It is tempting to conflate these two ideas, but they don’t directly affect each other in any way.

THE GAMBLERS FALLACY, FAIR COINS, and PROBABILITY EXAMPLES: The probability of a fair coin landing on tails twice in a row is 25% (50% chance on toss #1, plus 50% chance on toss #2; or one “two heads event” and three “not two heads events”; 1:3 or 25% total chance of getting two heads events in two tosses). However, despite this, each unique toss event has a 50% chance of producing heads; there is always one “heads event” for every “not heads AKA tails” event in a single toss. Betting against heads happening twice in a row before it occurs is a good bet, yet betting against an individual toss two consecutive times is a much different bet, it gives you exactly a 50% chance of being right each time. In other words, the probability of a single event never changes. Once the first toss has occurred, the second toss will have a 50% of occurring no matter what. There are lots of ways to explain this, and the page will do just that, but the mathematical truism never changes and cannot change. The belief that this is not true, that since it was black ten times in a row the next one is more likely to be red, or since it was tails ten times the next must be heads, is the gambler’s fallacy, it is the most common misconception in gambling. The key is understanding the difference between the likelihood something will happen multiple times in a row vs. the likelihood that the next event will produce specific results, they aren’t as related as they seem!

TIP: Odds and probability are used in both statistics and gambling. We use games of chance to explain the concepts on this page because they are easy to visualize. However, the core concepts expressed here are probably best applied to non-casino related ventures like weather, healthcare, politics, insurance, sales, big data, computing, or finance.

The Gambler’s Fallacy Explained

The Gambler’s Fallacy (Monte Carlo Fallacy or the Fallacy of the Maturity of Chances) is the name given to the mistaken belief that, in truly random events, the odds of something happening increase based on past events.

An example of Gambler’s Fallacy Belief: The coin landed on heads ten times in a row so the odds of it landing on tails have increased.

Likewise, the inverse Gambler’s Fallacy says is the belief that something happening more often indicates that the same result is more likely to occur in the next event.

An example Inverse Gambler’s Fallacy Belief: The coin landed on heads ten times in a row so the odds of it landing on heads have increased.

Both the Gambler’s Fallacy and the inverse gambler’s fallacy are incorrect. The probability of a true random independent event never changes.

FACT: If you had to pick one, the inverse Gambler’s Fallacy will be right more often. That is because games like roulette, for example, may have a slight inconsistency that results in the ball favoring a part of the wheel. It’s not something to bet on, but it’s worth mentioning that physical change can change the odds.

The Fair Coin Example

A coin flip is probably the simplest way to visual why the Gambler’s Fallacy and the statistics behind it:

Take a fair coin. Theoretically, a coin that lands on heads exactly 50% of the time and tails exactly 50% of the time.

Every time we flip the coin it has a 1 in 2 (or 1⁄2 ) chance of landing on heads or tails.

Say we flip a “fair coin” 20 times, and it lands on heads 20 times, the next flip has exactly a 1 in 2 chance of landing on heads.

Now instead we don’t flip the coin at all, but just calculate the probability of the coin landing on heads 21 times in a row. We get 1 in 2,097,152![4]

Before the coin is flipped the theoretical odds of getting the same result 21 times in a row are slim (1⁄2,097,152). Each time the coin is flipped the theoretical odds decrease exponentially all the way to the actual 1⁄2 odds at the last flip.

We can see that the actual odds are constant, but the theoretical odds change based on how many coin flips are left in a series.

FACT: The actual odds can be used to calculate theoretical odds. Knowing the probability of one flip is always 1⁄2 ,we can calculate two flips as 1⁄4, three flips is 1⁄8 the number increases exponentially until it goes well beyond 1⁄2,097,152.

Ignore Your Logic and Believe the Math

Logic might tell you that it is somehow more likely that a coin will land on tails or heads based on past outcomes, but this is simply the most pervasive of all gambling myths, the Gambler’s Fallacy. There is no such thing as an outcome being “overdue” in a truly random event. Past events have no memory.

The Monte Carlo Fallacy Origin Story

The use of the term Monte Carlo Fallacy originates from the most famous example of this phenomenon, which occurred at the Monte Carlo Casino on August 18th, 1913. On that night, the roulette wheel had landed on black many times in a row. People were sure that meant it had to land on red soon. It did, but not until 26 spins later by which time people had lost an enormous amount of money betting on black. Those who won on the 27 spin had another streak of “bad luck” when they assumed the roulette wheel would now switch to landing on red to “balance out” the past results. Of course, past results don’t affect future odds in a game of chance. So much money was lost due to this “Gambler’s Fallacy” that night that it is now known as the Monte Carlo Fallacy.[3]

FACT: The “quantum Monte Carlo algorithm” is a computer program that can calculate probability in a way that mimics a quantum computer. It uses the principles of Monte Carlo fallacy and the rule of large numbers as its basis to “determine” outcomes. And you thought we were just learning about gambling.[5]

Breaking the Rule and Clarification

The above information is always true for “true random” events. However, there is a hand full of important concepts that go along with the information:

Determinism and Pseudo-Random Numbers

Computers, humans, and the tools we build are all “deterministic“. This means they aren’t actually truly random, but instead their actions are defined by a preexisting set of values. This is why we use the theoretical “fair” coin example instead of debating the philosophical aspects of “randomness“.

While X slot machine may always have a 1 in 1,000,000 chance of hitting the jackpot, it (being electronic) uses a “pseudo random number” generator to control events. This means that, theoretically, one could backwards engineer the algorithm used in the slot machine to determine its outcome (in practice this doesn’t work due to logistics and the use of “true” random seed numbers derived from nature).

The same line of thinking works for humans and the machines they build. Theoretically, you can get a feel for how a certain craps player rolls dice and predict the outcome of their next role by studying their past behaviors and the outcome of the dice.

More realistically, you can focus on the practical human aspect and play a “skill” based game like poker. It’s reasonable to study other players to better predict their behaviors or to learn a strategy for a game.

Despite some of this being true theoretically, it doesn’t actually affect the way statistics works in regards to probability.

NOTE Explaining slot machine “Random Numbers” Generators: In reality things like slot machines and the lotto will use a random “seed” number. Some natural occurring phenomena, like radioactive decay, can be measured to get “true” random values. These values can be used as “seed” numbers making the pseudo random algorithms impossible to “determine”.

Cheating, Mechanical Inconsistencies, Betting Systems

Albert Einstein once said, “The only way to make money at a roulette wheel is to steal it when the dealer isn’t looking.” (NOTE: It’s likely that Einstein never said this, but it is still a good turn of phrase. The other way to win at roulette is to get lucky and then walk away.

Despite the fact that humans and machines are predictable, algorithms that make their theoretical predictability all but meaningless in a practical sense run the games of chance in a casino.

Mathematically speaking there is no betting system that can give you a house edge on games of chance like roulette. In most casino games the odds are against the player in both the short and long term. Barring cheating or inconsistencies in a machine, one shouldn’t expect a consistent edge over the house at any time.

The best you can hope for in games of “skill”, like blackjack, is to use a perfect strategy based on mathematics to increase your “edge”. Betting systems are only effective in skill based games where how you play can affect the odds (although without cheating, the house always wins in the long run, even with a perfect strategy).[1][2]

Theoretical Odds and Standard Deviation

The odds of things happening (like the lotto and slot machines) are theoretical odds. They are expected results over an infinite (or very large) number of plays. In a game of chance theoretical odds can at best give you a slight idea of the chances of something happening.

More than theoretical odds, in the short term you would be concerned with standard deviation (chances of an expected result happening or hit frequency). Consider a slot machine that has a 100% return rate (it theoretically is programed to pay a $1 for every dollar put in over the long term). The slot machine theoretically pays out one jackpot of one million dollars every one million spins and takes a dollar per spin. If you take that to mean that you can put a million dollars in and get a million dollars back, stay away from the casino. All this means is that on average, over an infinite amount of spins, it would payout a $1 for every $1 put in. In reality you could easily go millions of spins without a win. The “chances” of a win are one in a million each spin; so even after a million spins, you would still have a 1 in a million chance.

The above being said the more theoretical spins we consider, the closer the expected outcome will be to the theoretical odds. This is due to something the casino’s count on to make a profit “the rule of large numbers”.

FACT: In simple terms, standard deviation is a measure of how many sets of results vary on average from an expected result. This can be thought of as representing “hit frequency”, how often you can expect to win in the here and now. The lower the standard deviation, the less the results vary, and thus (for the purposes of gambling at least) the better the odds are in the short term. The better the theoretical odds, the better the odds are in the long term. The more events that are played out, the more the theoretical odds matter and the less the standard deviation matters.

Rule of Large Numbers

Theoretical odds tend to actualize over the long term due to the rule of large numbers which says “the more trials of a random process that occur, the closer the actual outcome will be to the theoretical outcome.”

Considering in the casino the house always has “the edge” this means that the more you play at a casino with a consistent bet, the higher the chances of you losing money over the long term are (the higher the chance of the theoretical odds materializing against you). Let’s look at a “fair coin” again for an example.

A given coin flip has 1 in 2 odds, but over a small number of flips (a hundred flips for example) there is a good chance in reality the coin won’t land on heads exactly 50 times and tails exactly 50 times. However, the more times we flip the coin, the more likely it will be that our theoretical probability of 1⁄2 will actualize. The more we flip the coin, the more likely we will be to have it land on heads or tails 50% of the time.

Odds in Statistics Versus Odds and Gambling

In common terms probability and odds are used interchangeably, and that is fairly true in math too, however in gambling odds mean something different. In statistics probability is the percentage chance that an outcome will happen and odds are just another way of expressing that chance. In gambling, the term “odds” is used to express probability and it also references a payout. Example: 4 – 1 (or 4 to 1 or 4:1) odds means the game pays $4 for every $1 bet.

Conclusion

As attractive as it is to think, the actual odds of an outcome aren’t determined by past events. The probability of an outcome exists independently of theoretical odds. Theoretical odds only actualize over a large number of events, and there is little we can do outside of cheating to truly affect games of chance in the short term.